Download BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES 1

BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
NOAM GREENBERG AND ANDRÉ NIES
Abstract. We show that the class of strongly jump-traceable c.e. sets can be
characterised as those which have sufficiently slow enumerations so they obey a
class of well-behaved cost function, called benign. This characterisation implies
the containment of the class of strongly jump-traceable c.e. Turing degrees in
a number of lowness classes, in particular the classes of the degrees which lie
below incomplete random degrees, indeed all LR-hard random degrees, and all
ω-c.e. random degrees. The last result implies recent results of Diamondstone’s
and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use
of algorithmic randomness in the study of the c.e. Turing degrees.
1. Introduction
K-triviality has become central for the investigation of algorithmic randomness.
This property of a set A ∈ 2ω expresses that A is as far from random is possible,
in that its initial segments are as compressible as possible: for all n, K(A n ) ≤+
K(n).1 The robustness of this class is expressed by its coincidence with several
notions indicating that the set is computationally feeble (Nies; Hirschfeldt and
Nies; Hirschfeldt, Nies and Stephan; see [16, 9, 15]):
• lowness for Martin-Löf randomness: as an oracle, the set A ∈ 2ω cannot
detect any patterns in a Martin-Löf random set.
• lowness for K: as an oracle, the set A cannot compress any strings beyond
what can be done computably.
• base for Martin-Löf randomness: A is so feeble that some A-random set
can compute A.
The key for this equivalence is the notion of cost functions and obeying them. The
by-now standard constructions of a promptly simple K-trivial set ([6]) shows that
a set which is low for Martin-Löf randomness ([11]) has similar dynamic properties
as a set which is low for K (Mučnik, see [2]). The requirements, which want
to enumerate numbers into the set A which is being built, are restrained from
doing so not by discrete negative requirements, such as in the standard Friedberg
construction of a low set, but by a cost function, which has a more continuous
nature. It is this resemblance between the constructions of “typical” representatives
in the classes mentioned above, which was the seed for the proofs of equivalence of
these notions. Technically, this is summarised in the Main Lemma of [15, Section
5.5], which indeed yields the hard implication. An application of that Main Lemma
characterizes the class of K-trivial sets by the standard cost function cK : a ∆02 set
A is K-trivial if and only if there is a computable approximation hAs i of A such
Both authors were partially supported by the Marsden Fund of New Zealand.
1Here K denotes prefix-free Kolmogorov complexity.
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NOAM GREENBERG AND ANDRÉ NIES
that
X
cK (x, s)[[x is least such that As (x) 6= As+1 (x)]]
s<ω
P
is finite (here cK (x, s) = x<y<s 2−Ks (y) ). In some sense, the canonical way to
construct K-trivial sets is the only way to construct these sets.
All known characterisations of the class of K-trivial sets involve an analytic
component such as Lebesgue measure or prefix-free Kolmogorov complexity. A
still standing question is whether this class can be defined using purely combinatorial tools, as used in computability theory outside its interaction with algorithmic
randomness. A one-time candidate was the class of strongly jump-traceable sets.
Traceability was introduced into computability theory by Terwijn and Zambella
[20] for their study of another lowness notion, that of lowness for Schnorr randomness; a variant was also used by Ishmukhametov [10] in his study of strong minimal
covers in the Turing degrees. A third variant, jump-traceability, was introduced by
Nies in [17]. The strong version of jump-traceability was defined by Figueira, Nies
and Stephan [7]. They showed that a non-computable strongly jump-traceable c.e.
set exists. For the formal definitions, recall that a c.e. trace for a partial function ψ
is a uniformly c.e. sequence hTx i of finite sets such that for all x ∈ dom ψ we have
ψ(x) ∈ Tx . An order function is a computable, non-decreasing and unbounded
function h : ω → ω \ {0}. A c.e. trace hTx i is bounded by an order function h if
for all x, |Tx | ≤ h(x). Finally, a set A is strongly jump-traceable if for every order
function h, every partial function ψ : ω → ω which is partial computable in A has
a c.e. trace which is bounded by h.
In [3], Cholak, Downey and Greenberg showed that the attempt to define Ktriviality using strong jump-traceability fails, but that in fact, restricted to the c.e.
degrees, the strongly jump-traceable degrees form a proper sub-ideal of the ideal
of K-trivial degrees. This was the first known example of such an ideal. Several
questions remained open:
(1) How does the ideal of strongly jump-traceable c.e. sets relate to other ideals
and classes of degrees, most of which are known to be contained in the
K-trivial degrees but are not yet known to be distinct from the ideal of Ktrivial degrees? Here we mostly think of classes derived from algorithmic
randomess, such as the collection of degrees which are bounded by an incomplete random degree. Cholak, Downey and Greenberg showed in their
paper that the strongly jump-traceable degrees are all ML-non-coverable,
another example for such a class, but no further examples were known.
(2) Can the strongly jump-traceable sets be characterised by cost functions?
Related to that is the question of the complexity of this ideal. Can traceability for some fixed order ensure strong jump-traceability? Is this ideal a
Σ03 ideal, like the K-trivial ideal, or is it more complicated?
(3) What is the status of non-c.e. strongly jump-traceable sets? The notion
of K-triviality is inherently enumerable; the cost function characterisation
implies that every K-trivial set is computable from a c.e. one. Does the
same hold for strong jump-traceability?
(4) Are there other characterisations of the strongly jump-traceable sets, which
would indicate that this notion is robust?
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
3
(5) Are there other natural ideals between the strongly jump-traceable degrees and the K-trivial degrees? Are there natural proper sub-ideals of the
strongly jump-traceable degrees?
The main results. We answer the first two questions. Indeed it is the solution for
the second question which gives a unified approach to one kind of “box-promotion”
constructions, two examples of which were first given in [3], which imply our solutions to the first question. We give a few definitions following [15].
A monotone cost function is a computable function c which associates with every
number x < ω and stage s < ω a “cost” c(x, s), a non-negative rational number, of
changing the approximation As (x) for membership of x in A at stage s. We require
that for each x, the sequence hc(x, s)is<ω is non-decreasing, and converges to a
limit c(x); we require that the limit cost function c(x) is non-increasing with x, and
indeed, that for any fixed stage s, the cost hc(x, s)ix<ω at stage s is non-increasing.
We say that a computable approximation hAs i of a ∆02 set A obeys a cost function
c if the sum
X
c(x, s)[[x least such that As (x) 6= As+1 (x)]]
s<ω
is finite. We say that a ∆02 set obeys a cost function c if there is some computable
approximation hAs i for A which obeys c. In this terminology, Nies’s result above
is that a ∆02 set is K-trivial if and only if it obeys the standard cost function cK .
We note, by the way, that if A is a c.e. set which obeys a cost function c, then A
has a computable enumeration which obeys c.
We usually require our cost functions to satisfy the limit condition limx→∞ c(x) =
0, where again c(x) = lims c(x, s). In this paper we introduce a class of cost function
that satisfy the limit condition in a restrained and predictable manner.
Definition 1.1. A (monotone) cost function c is benign if there is a computable
function g such that for every positive rational ε, g(ε) bounds the size of any
collection I of pairwise disjoint intervals of natural numbers such that for all [x, s) ∈
I we have c(x, s) ≥ ε.
For example, the standard cost function cK is benign: for any ε > 0, any I as
in the definition cannot have size greater than 1/ε, because the witnesses, in the
universal machine, for cK (x, s) ≥ ε for [x, s) ∈ I must all be distinct; this is because
the intervals in I are disjoint.
Here is another way to understand the definition of benignity. Let ε > 0. Set
y0ε = 0, and if ykε is defined, and there is some s such that c(ykε , s) ≥ ε, then set
ε
yk+1
to be the least such s. If c satisfies the limit condition limx c(x) = 0, then
this process has to halt after finitely many iterations. Then c is benign iff there is
a computable (in ε) bound on the number of iterations of this process.
Our main theorem settles the first part of Question (2) above.
Theorem 1.2. A c.e. set A is strongly jump-traceable if and only if it obeys all
benign cost functions.
Since cK is benign, this implies the main result of [3], that every strongly jumptraceable c.e. set A is K-trivial.
As for the second part of Question (2), we show the following.
Theorem 1.3. For any benign cost function c, there is a c.e. set A which obeys c
and is not strongly jump-traceable.
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NOAM GREENBERG AND ANDRÉ NIES
In light of Theorem 1.2, this says that for every benign cost function c there
is another, more stringent, and yet still benign, cost function d, such that there
is a c.e. set which obeys c but not d. Theorem 1.3 implies Cholak, Downey and
Greenberg’s result from [3] that the ideal of strongly jump-traceable degrees is
strictly contained in the ideal of K-trivial degrees.
Together with Proposition 2.2, which is a uniform version of the proof of the
easy direction of Theorem 1.2, Theorem 1.3 implies Ng’s result [12] that no single
order function h can ensure strong jump-traceability. Indeed, in the same paper,
Ng went on to show that the index-set for the collection of strongly jump-traceable
c.e. sets is Π04 -complete, and so certainly is not Σ03 .
Applying Theorem 1.2. The main theorem allows us to unify constructions
which show that the strongly jump-traceable degrees are a subclass of most of the
subclasses of the K-trivial degrees which are considered in the study of algorithmic
randomness.
Recall that Y ≤LR X if every X-random set is Y -random. The class LRH of sets
X such that ∅0 ≤LR X, or LR-hard sets, is also known as the class of “high for
random” sets. A set is LR-hard if and only if it is (uniformly) almost everywhere
dominating (Kjos-Hanssen, Miller, Solomon; see [18, 15]).
Theorem 1.4. Every strongly jump-traceable c.e. set is computable from any LRhard random set.
As an immediate corollary we see that in the c.e. degrees, the collection of
strongly jump-traceable degrees is contained in the collection of degrees which are
bounded by incomplete random degrees. We remind the reader that it is still unknown if this latter collection coincides with the collection of K-trivial degrees.
Hirschfeldt showed that if A is an incomplete c.e. set, X is an incomplete random
set, and ∅0 ≤T A ⊕ X, then X is LR-hard (see [15, Theorem 8.5.15]). Hence
Theorem 1.4 implies the result from [3], that no strongly jump-traceable c.e. set is
M L-cuppable. Again, it is unknown whether the class of M L-non-cuppable sets
coincides with the K-trivials.
The authors were surprised to discover the following result.
Theorem 1.5. Every strongly jump-traceable c.e. set is computable from any ω-c.e.
random set.
In contrast with the situation regarding the LR-hard random sets, we can show
(Theorem 5.3) that there is a K-trivial set which is not computable in some ω-c.e.
random set. Indeed, recent research by the authors together with D. Hirschfeldt
shows the coincidence of the c.e. strongly jump-traceable sets and the sets which
are computable from every ω-c.e. random sets.
Theorem 1.5 can be improved as follows. In [8], the authors define a binary
relation which is a very strong version of weak truth-table reducibility: Y ≤T (tu) X
(Y is reducible to X with tiny use) if for every order function h, there is a reduction
of X to Y whose use function is bounded by h. By making our cost functions
stringent, we can in fact show that if A is c.e. and strongly jump-traceable, and X
is an ω-c.e. random set, then A is reducible to X with tiny use (Proposition 5.2).
The analogy between Theorem 1.4 and Theorem 1.5 leads to the following definition. For a class C of sets, we let C♦ be the class of all c.e. sets A which are
computable in every random set in C. Thus, these two theorems can be stated as
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
5
the inclusions
SJT ⊆ LRH♦
and
SJT ⊆ (ω-c.e.)♦ ,
where SJT denotes the collection of strongly jump-traceable c.e. sets.
We recall that Hirschfeldt and Miller (see [19] or [15]) showed, via a cost function
construction, that if C is a null Σ03 class, then C♦ contains a promptly simple c.e.
set. Thus the content of Theorems 1.4 and 1.5 is that if for a given C, Hirschfeldt
and Miller’s construction happens to use a benign cost function, then every strongly
jump-traceable c.e. set can be viewed as being produced by their construction.
Applying randomness in degree theory. The class (ω-c.e.)♦ has an unexpected
application. Recall that a set B is superlow if B 0 ≤wtt ∅0 . A problem in c.e. degree
theory, which turned out to be quite difficult to solve, was whether superlow cuppability coincided with low cuppability (which in turn was shown to be equivalent
to having a promptly simple degree in the classic [1]). This question was recently
settled in the negative by Diamondstone [5]. In parallel, Ng [13] showed that in
analogy with the almost deep degree of [4], there is a c.e. degree which joins every
superlow c.e. degree to a superlow degree; he called such degrees almost superdeep.
We show in Section 5, using the class (ω-c.e.)♦ , that every strongly jump-traceable
c.e. set is almost superdeep, thus extending the results of both Diamondstone and
Ng.
The remaining questions. To end this introduction, we discuss the status of
Questions (3)-(5) from Page 2. In yet unpublished work, Downey and Greenberg
showed that in contrast with the jump-traceable sets, every strongly jump-traceable
set is ∆02 . Theorem 1.2 seems to indicate that like K-triviality, strong jumptraceability is a notion which is inherently enumerable. Downey and Greenberg
currently conjecture that every strongly jump-tracable set is computable in a c.e.
one; this would imply that all the results in this paper extend to all strongly jumptraceable sets. It seems likely that the characterisation of strong jump-traceability
in terms of cost functions will play an important role in the verification of this
conjecture.
For the fourth question, recent preliminary results of the authors together with
Hirschfeldt indicate that the class of strongly jump-traceable degrees is indeed robust, as it may coincide with some of the “diamond classes” defined above, such
as (ω-c.e.)♦ , Superlow♦ and Superhigh♦ . Again these results use benign cost functions in a fundamental way. However, no natural classes which lie strictly between
the strongly jump-traceable and the K-trivial degrees have yet been found. Ng
[14] has defined and investigated an ideal which is strictly contained in the strongly
jump-traceable degrees. This ideal is obtained by partially relativising strong jumptraceability to all c.e. sets, and seems to be Π05 -complete. In contrast with the
strongly jump-traceable degrees, the degrees in this small ideal cannot be promptly
simple. Not much is currently known beyond these results.
2. Proof the main theorem
In this section we prove Theorem 1.2. We first prove the easy direction: that
if A is a c.e. set which obeys every benign cost function, then A is strongly jumptraceable. This is implied by the following Proposition 2.1. Recall that for any
6
NOAM GREENBERG AND ANDRÉ NIES
set A, J A denotes a universal A-partial computable function; to show that a set
is strongly jump-traceable, it is sufficient to show that for every order function h,
J A has a c.e. trace which is bounded by h. Note that if A obeys every benign
cost function, then it obeys cK , and so is K-trivial. It follows ([17],[16]) that A
is jump-traceable: every A-partial computable function has a c.e. trace which is
bounded by some order function. Hence it is sufficient to prove the following.
Proposition 2.1. Let A be a c.e., jump-traceable set, and let h be an order function. Then there is a benign cost function c such that if A obeys c, then J A has a
c.e. trace which is bounded by h.
Proof. Since A is c.e., tracing J A (n) is equivalent to tracing the correct J A (n)
computation. In other words, let ψ A (n) be the stage at which J A (n) converges
with an A-correct computation. Since A is jump-traceable, there is a c.e. trace
hSn i for ψ A which is bounded by some order function g.
Suppose that J Ar (n)↓. We say that this computation is certified if there is some
t < r such that Ar u = At u (where u is the use of the computation) and such that
t ∈ Sn at stage r. We want to make sure that the cost of all x < u at stage r is at
least 1/h(n). Hence we let
1
Ar
: ∃ r ≤ s J (n) is certified, with use u > x .
c(x, s) = max
h(n)
Note that this definition indeed makes c monotone.
We first argue that c is benign. Let ε > 0 and suppose that I is a set of pairwise
disjoint intervals of natural numbers such that for all [x, s) ∈ I, c(x, s) ≥ ε. Find
some n∗ such that h(n∗ ) > 1/ε. Let [x, s) ∈ I. Then there is some n < n∗ and
r ≤ s such that J Ar (n), with use u > x, is certified. Say that t < r witnesses that
J Ar (n) is certified. The key is that t ∈ (x, s) (as x < u < t < r ≤ s), and so, since
the intervals in I are pairwise disjoint, and t ∈ Sn , we have
X
X
|I| ≤
|Sn | ≤
g(n).
n<n∗
n<n∗
Since n∗ is obtained effectively from ε and g is computable, this bound on |I| is
also effective.
D E
bs be a computable enumeration of A such
Now suppose that A obeys c. Let A
that
X
bs (x) 6= A
bs+1 (x)]] < 1.
c(x, s)[[x least such that A
s<ω
Enumerate a trace hTn i for J A as follows: enumerate J As (n) into Tn at stage s if
bs u , where u is the use of the computation
there is some r < s such that Ar u = A
bs
A
J (n), and this computation gets certified at stage r.
Let n < ω and let s0 < s1 < · · · < s|Tn |−1 be the stages at which we enumerate
b
numbers into Tn ; say that the computation J Ask (n) gets certified at stage rk < sk .
Let uk be the use of that computation. For each k < |Tn | − 1 there is some stage
bw u 6= A
bw +1 u , as by stage sk+1 , the computation
wk ∈ [rk , rk+1 ) such that A
k
k
k
k
b
J Ask (n) is injured by some number below uk entering A. By design,
b
c(uk − 1, wk ) ≥ c(uk − 1, sk ) ≥ 1/h(n).
Hence |Tn | − 1 ≤ h(n). Now replacing h by h − 1 completes the proof.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
7
In fact, the proposition can be uniformised:
Proposition 2.2. For every order function h there is a benign cost function c such
that for any c.e. set A which obeys c, J A has a c.e. trace which is bounded by h.
The proof of Proposition
trace for an order
j√ k 2.2 uses the notion of a universal
h . There is an effective listing S 1 , S 2 , S 3 , . . . of all c.e.
function h. Let e
h=
S
traces which are bounded by e
h. Let Tn =
S e . Then hTn i is a c.e. trace
e
e<h(n)
n
which is bounded by h, and for every partial function ψ, if ψ has a c.e. trace which
is bounded by e
h, then for almost all n ∈ dom ψ, ψ(n) ∈ Tn , so hTn i almost traces
ψ. We call hTn i a universal trace for h.
Proof of Proposition 2.2. By [17], there is an order function ge such that for every Ktrivial set A, every A-partial computable function has a c.e. trace which is bounded
by ge. Let g = ge2 , and let hTn i be a universal trace for g.
Now let c be the cost function which is obtained by running the proof of Proposition 2.1 using the c.e. trace hTn i and all possible c.e. oracles A. Namely, we say
that a computation J We,r (n) is certified if We,r u = We,t u where u is the use of
the computation, and t ∈ Tn at stage r. We let
1
: ∃ e, r ≤ s J We,r (n) is certified, with use u > x .
c(x, s) = max
h(n)
Again this definition makes c monotone, and the argument in the proof of Proposition 2.1 shows that c is benign.
Since both c and cK are benign, so is the cost function c + cK . If A is a c.e. set
which obeys c + cK , then it obeys both c and cK . It follows that A is K-trivial,
so the converging time function ψ A for J A has a c.e. trace bounded by ge; so ψ A
is almost traced by hTn i. The last paragraph of the proof of Proposition 2.1 now
shows that J A is almost traced by a c.e. trace which is bounded by h. Of course a
finite modification gives a full trace.
We now turn to the proof of the harder direction of Theorem 1.2: we show that
if A is c.e. and strongly jump-traceable, then A obeys every benign cost function.
The argument is a generalisation of the box-promotion method proof from [3] which
shows that every strongly jump-traceable c.e. set A is K-trivial. Indeed, we prove
a converse of the uniform Proposition 2.2:
Proposition 2.3. For any benign cost function c, there is an order function h such
that for any c.e. set A, if J A has a c.e. trace bounded by h, then A obeys c.
Instead of using the recursion theorem as in [3, 15], we rely on universal traces.
We first note that for every order function h0 there is an order function h such
that for any set X, if J X has a c.e. trace bounded by h, then every X-partial
computable function has a c.e. trace which is bounded by h0 (we say that X is
h0 -jump-traceable). This is because every X-partial computable function is coded
in the jump function, and we can uniformly limit the rate of growth of the functions
which give the coding locations. So if hTn i is a universal trace for e
h = (h0 )2 , then
X
if J has a c.e. trace bounded by h, then every X-partial computable function is
almost traced by hTn i.
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NOAM GREENBERG AND ANDRÉ NIES
The general idea of any box-promotion construction with c.e. oracle A is to
certify certain A-configurations up to varying degrees of certainty. To this end,
we define an A-partial computable function ΨA ; to certify As u we define, at
stage s, ΨAs (z) for various z with use u and output s; the configuration is then
certified at a later stage t if At u = As u and s ∈ Tz at stage t. The degree of
certainty this certification gives us depends on the bound e
h(z) we have for the size
of Tz ; we know that we cannot make more than e
h(z) − 1 many mistakes. So if, for
example, e
h(z) = 1, and As u is certified at stage t, then we know that A u = As u .
Unfortunately, though, for almost all zDwe E
have e
h(z) > 1.
b
Specifically, to find an enumeration As of A which obeys c, we want to speed
up a given enumeration hAs i of A and only accept sufficiently certified configubs (x) changes on some x such that
rations of A. To ensure obedience to c, if A
−n
c(x, s) ≥ 2 , say, then we want to make progress, in the sense that the previous
version of A x+1 was certified by some Tz such that e
h(z) ≤ n. The idea is to ensure
that because of this limit on |Tx |, this won’t happen more than n times. Hence the
sum
X
bs (x) 6= A
bs+1 (x)]]
c(x, s)[[x is least such that A
s<ω
will be bounded by
X
n2−n
n<ω
which is finite.
The part of the construction which deals with those x’s for which c(x, s) ≥ 2−n ,
call it requirement Rn , may ignore those x’s for which c(x, s) ≥ 2n−1 , as these need
to be certified in even stronger “boxes” Tz . All of these certification processes need
to work in concert; in general, at a given stage s, we will have u1 < u2 < u3 < . . .
such that As u1 has to be certified with strength 2−1 (by R1 ), As u2 has to be
certified with strength 2−2 (by R2 ), etc. The problem is that not every ΨA (z) is
traced by Tz ; there are finitely many exceptions. Hence for every d < ω, a version
of the construction indexed by d will guess that ΨA (z) is traced by Tz for each z
such that e
h(z) ≥ d. Almost all versions will be successful. To keep the various
versions from interacting, each version will control its own (infinite) collection of
boxes Tz . That is, for every z, only one version of the construction will attempt to
make definitions of ΨA (z).
A common feature of all box-promotion constructions, is that certification takes
place along a whole block of boxes Tz which together form a “meta-box”. The
point is that to ensure that Rn only certifies n − 1 many wrong initial segments of
A, we need each failure to correspond to an enumeration into the same Tz . On the
other hand, if a correct initial segment is tested on some Tz , then this z is never
again available for testing other, longer initial segments of A. The idea is that if
one meta-box B used by Rn is promoted (by some s ∈ Tz for all z ∈ I discovered
to be wrong), then we break B up into many sub-boxes, and so on. The fact that c
is benign, witnessed by a computable bound function g, allows us to set in advance
the size of the necessary meta-boxes, thus making e
h computable. A meta-box for
Rn can be broken up at most n times, so the necessary size for an original Rn
n+1
meta-box is (g(2−n ))
.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
9
Definition of e
h and the initial meta-boxes. Let hIn in≥1 be consecutive, pairwise
n+1
disjoint intervals of ω such that |In | = n(g(2−n ))
. For all z ∈ In , let e
h(z) = n.
n+1
Next, we split each In into intervals In1 , In2 , . . . , Inn , each of size (g(2−n ))
. The
interval Ind will be used by the d-version of Rn , which we denote by Rnd . So we set
d
Bn,0
, the initial meta-box for Rnd , to be Ind .
d
At any stage s, Bn,s
will be an interval of ω whose size is a power of g(2−n ). For
−n
d
d
d
k ∈ {1, 2, . . . , g(2 )}, we let Bn,s
(k) be the k th subinterval of Bn,s
of Bn,s
of size
d
−n
|Bn,s |/g(2 ).
d-stages and certification. As mentioned above, the d-version of the construction
guesses that for all n ≥ d, for all z ∈ Ind , if ΨA (z)↓ then ΨA (z) ∈ Tz . The d-stages
sdi are defined by recursion; these are the stages at which this guess looks correct.
We let sd0 = 0. Given sdi , let sdi+1 be the least stage s > sdi such that for every
n ∈ [d, i], for all z ∈ Ind , either ΦAs (z) ↑ or ΦAs (z) ∈ Tz,s . We ensure that the
d-version of the construction only makes definitions of ΦA at d-stages. Hence, if
the d-version of the construction guesses correctly, there will be infinitely many
d-stages.
For a d-stage s = sdi+1 , let s̄ = sdi be the previous d-stage. We say that As u is
certified if
As u = As̄ u .
Definition of Φ. Fix d < ω and n ≥ d. Let Di > nEand s = sdi . We describe the
−n
action of Rnd at stage s. Recall the sequence yk2
from the introduction, which
n
we rename hykn i: y0n = 0, and if ykn is defined, then yk+1
is the least s such that
n
−n
n
c(yk , s) ≥ 2 , if such a stage s exists; otherwise, yk+1 is not defined. At a stage s
n
we can compute all ykn for which ykn ≤ s. We know that yg(2
−n ) is not defined.
n
The aim is that by the end of stage s, if k ≥ 1, yk ≤ s̄ is defined and As ykn is
d
(k): we say that
certified, then we will have ΦAs (z)↓ with use ykn for all z ∈ Bn,s
d
As ykn is tested in Bn,s (k). The inductive hypothesis on the construction is that
this indeed holds for all such k at the end of stage s̄, whereas if ykn is not defined at
d
stage s̄, or it is defined but As̄ ykn is not certified, then for all z ∈ Bn,s̄
(k) we have
ΦAs̄ (z)↑.
First, to see if Rnd can make progress, we check if there is some k ≥ 1 such that
d
As̄ ykn was tested in Bn,s̄
(k), and such that
As ykn 6= As̄ ykn .
If so, then Rnd can promote its meta-box Bnd : We reset
d
d
Bn,s
= Bn,s̄
(k),
d
where k is the least witness. We note that in this case, for every z ∈ Bn,s
we have,
As
before we make any new definitions, Φ (z)↑, because at stage s̄ we have ΦAs̄ (z)↓
with use ykn . Hence we can define ΦAs (z) for such z as we like: for all l ∈ [1, k),
d
As yln is certified, and so for all z ∈ Bn,s
(l) we define ΦAs (z) = s with use yln .
d
d
Now if Rnd does not promote its meta-box at stage s, then Bn,s
= Bn,s̄
; for all
n
n
k ≥ 1 such that As̄ yk was tested at stage s̄, As yk is still certified, and is still
d
d
tested in Bn,s
(k) = Bn,s̄
(k). If there are k such that ykn ≤ s and As ykn is certified,
10
NOAM GREENBERG AND ANDRÉ NIES
d
d
(k) = Bn,s̄
(k) we have
but As̄ ykn was not tested at stage s̄, then for all z ∈ Bn,s
As̄
As
n
Φ (z)↑, so we can define Φ (z) = s with use yk for all such z.
D E
bs of s and
This ends the construction. Before we define the enumeration A
show that the enumeration obeys c, we need to make sure that the construction is
consistent, in that the instructions can always be carried out. We can easily verify
that the inductive hypothesis holds and the end of stage s: if ykn < s and As ykn is
d
d
certified, then it is tested in Bn,s
(k); otherwise, for all z ∈ Bn,s
(k) we have ΦAs (z)↑.
Another issue is to verify that each requirement Rnd can always promote its metad
box Bnd when that is required, that is, it can divide Bn,s
(k) to at least g(2−n ) many
d
sub-intervals. This follows from the size of the original meta-box Bn,0
= Ind and
the following lemma:
Lemma 2.4. The procedure Rnd does not promote its meta-box more than n times.
Proof. Let r < t be two stages at which Rnd promotes its box. Note for all s ≥ r, for
d
d
is redefined at stage r, we
(z), if ΦAs (z)↓ then ΦAs (z) ≥ r: when Bn,r
all z ∈ Bn,s
Ar
have Φ (z) undefined, and all new definitions, at stage r or afterwards, are made
d
d
= Bn,
with the value being the stage number. Let k be such that Bn,t
t̄ (k). By the
A
t̄
conditions for promotion, we have At̄ ykn certified and tested, so Φ (z) ∈ Tz,t (by
the definition of a d-stage, which t is). Hence there is a number s ∈ [r, t) in Tz for
all such z.
Also, if t is the first stage at which Bnd is promoted, then the same argument
d
.
shows that there is a number smaller than t in Tz for all z ∈ Bn,t
d
The meta-boxes are nested, so if Bn were promoted n + 1 times, say for the
d
. This
n + 1st time at stage s, we’d have n + 1 distinct numbers in Tz for all z ∈ Bn,s
d
d
d
e
contradicts the fact that Bn,s ⊂ In and for all z ∈ In we have n = h(z) ≥ |Tz |. D E
bs and show that this enumeration of A obeys c.
We turn to define A
Fix some d such that for all n ≥ d, for all z ∈ Ind , if ΦA (z)↓ then ΦA (z) ∈ Tz . So
there are infinitely many d-stages. From now, we drop the superscript d from Rnd ,
d
d-stage, sdi , Bn,s
, etc.
By recursion we define a sub-sequence of stages. Let q(0) = 0, and given q(r),
let q(r + 1) be the least stage s greater than q(r) at which As q(r) is certified.
br = Aq(r+2) r . For all r, let xr be the least x such that
For all r < ω, let A
b
b
Ar−1 (x) 6= Ar (x) (so xr < r). Let nr be the unique n such that
2−n ≤ c(xr , r) < 2−(n−1) .
D E
br obeys c is equivalent to showing that
Hence, showing that A
X
2−nr
r
is finite.
Lemma 2.5. For any r, there is some stage s ∈ (q(r + 1), q(r + 2)] at which Rnr
promotes its meta-box.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
11
Proof. Let n = nr and x = xr . Let k be the greatest such that ykn is defined and
ykn ≤ r. We have x < ykn , for otherwise, by monotonicity of c, we’d have
c(ykn , r) ≥ c(x, r) ≥ 2−n
n
is defined and is not greater than r.
which would imply that yk+1
n
Hence yk ≤ r ≤ q(r). The choice of x and the fact that x < ykn shows that
Aq(r+2) ykn 6= Aq(r+1) ykn .
However, by the definition of q(r + 1) and the fact that ykn ≤ q(r), Aq(r+1) ykn is
certified, so is tested on Bn,q(r+1) (k) at stage q(r + 1). We get a change on As ykn
by stage q(r + 2), so if s is the least stage beyond q(r + 1) at which As ykn is not
certified, then s ≤ q(r + 2) and Rn promotes its meta-box at stage s.
It follows that for all n,
{r : nr = n}
has size at most n, and so
X
r
2−nr ≤
X
n2−n
n
which is finite. This completes the proof of Proposition 2.3.
3. No single benign cost function suffices for strong
jump-traceability
In this section we prove Theorem 1.3: if c is a benign cost function, then there
is some c.e. set A which obeys c but is not strongly jump-traceable.
Let g be a computable bound function which witnesses that c is benign. In this
construction, for notational convenience, we replace g by g + 1, so g(ε) is strictly
greater than the number of pairwise disjoint intervals [x, s) such that c(x, s) ≥ ε.
To prove the theorem, we enumerate a c.e. set A; the enumeration hAs i which
we define will obey c. To ensure that A is not strongly jump-traceable, we design
an order function h and build a functional Ψ; we meet the requirements Re , which
say that the eth c.e. trace hSxe i with bound h does not trace ΨA . The idea is that
Re will work with potential witnesses in some interval Ie of natural numbers; we
will define h(x) = e for all x ∈ Ie , and so Re will want to change the value of ΨA (x)
for some x ∈ Ie at least e times. To ensure that hAs i obeys c, we sometimes need to
abandon a witness, because the cost of redefining ΨA (x), by enumerating the use
of the computation into A, becomes too large. It is the fact that c is benign, that
allows us to calculate in advance the total number of possible such abandonments
Re may need to concede, and so a bound on the size of Ie ; which is, of course,
necessary, since we need to make h computable.
To make the situation clear, we consider the first few requirements. At attempt
to meet R1 would have a witness x ∈ I1 for which we first define, at some stage s0 ,
ΨA (x) = s0 with use s0 + 1. At a later stage s1 , s0 appears in Sx1 , and we want
to enumerate s0 into A and redefine ΨA (x) = s1 , meeting the requirements since
|Sx1 | ≤ 1. If the cost c(s0 , s1 ) is greater than the quota, say 1/2, allocated to R1 ,
then we need to abandon x and start afresh with a new witness. This can happen
only fewer than g(1/2) many times, so we need |I1 | ≥ g(1/2).
Now consider R2 . The process is similar, except that if x is not abandoned at
stage s1 , then s1 may still appear in Sx2 at a yet later stage s2 , at which point we
want to enumerate s1 into A. We are now in a double bind, because enumerating
12
NOAM GREENBERG AND ANDRÉ NIES
s0 into A at stage s1 has already cost R2 the amount of c(s0 , s1 ), which was smaller
than R2 ’s quota (say another 1/2), but yet positive. If c(s1 , s2 ) is greater than
what’s left to spend (1/2 − c(s0 , s1 )), then we need to abandon x and start with
a fresh witness, with a net loss of c(s0 , s1 ) for R2 and no gain whatsoever. The
strategy is to take into consideration all possible such failures and “spread out
the investment”. Instead of being willing to spend it all each time, R2 declares
a quantity of 1/4 which is reserved to spending at a stage like s2 , i.e., when it is
ready to meet the requirement. This means that it may abandon the witness at
the stage s2 fewer than g(1/4) many times. Between such abandonments, it may
spend one unreturned cost at a stage s1 ; so the amount it is willing to spend at a
stage s1 should be no more than 1/4g(1/4). So between abandoning witnesses at
an s2 stage, we may abandon fewer than
!
1
g
4g 14
witnesses. It follows that we need
1
|I2 | ≥ g
g
4
!
1
4g
1
4
.
In general, Re ’s total capital allotment is e2−e , and it is willing to overall spend
2 at each level – stages s1 , s2 , . . . , se , or as we name them from now, by procedures
P1e , P2e , . . . , Pee .
e
Each procedure Pke calls a procedure Pk−1
and expects it to return, at some
stage sk , with a witness x and some sk−1 such that ΨA (x) = sk−1 with use sk−1 ,
and such that |Sxe | ≥ k at stage sk . It will then either violate that computation
by enumerating sk−1 into A, redefining ΨA (x) = sk with use sk + 1, wait for sk
e
to appear in Sxe , so that |Sxe | ≥ k + 1 and Pke can return to the procedure Pk+1
which called it. If k = e then the requirement is met. Or, if the cost csk (sk−1 ) of
enumerating sk−1 at stage sk is too big, bigger than a threshold δke , then x gets
e
cancelled, and a new run of Pk−1
is called.
We calculate the necessary thresholds δke and a bound nek on the total number
of times a procedure Pke can be called. We call Pee once, so let nee = 1. Hence
δee = 2−e , and nee−1 = g(δee ). Inductively, given nek for k > 0, we set
−e
δke =
2−e
nek
and
We require |Ie | =
ne0 .
nek−1 = nek g (δke ) .
This defines h.
We now describe the action of each procedure.
A procedure P0e , called at some stage s0 , chooses a fresh x ∈ Ie , defines ΨA (x) =
s0 with use s0 + 1, and waits for s0 to appear in Sxe . When this happens, the
procedure returns, with output x and s0 .
e
A procedure Pke , for 1 ≤ k ≤ e, calls a procedure Pk−1
. When that procedure
A
returns at stage sk , with a witness x such that Ψ (x) = sk−1 with use sk−1 + 1,
we compare c(sk−1 , sk ) and δke :
e
• If c(sk−1 , sk ) > δke , then we cancel x, and call a new run of Pk−1
.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
13
• Otherwise, we enumerate sk−1 into A, and redefine ΨA (x) = sk with use
sk + 1. We wait for sk to show up in Sxe . When this happens, the procedure
returns, with the witness x.
Note that we don’t want the different requirements Re to interfere with each
other, so when a procedure Pke defines ΨA (sk ) with use sk + 1, while waiting for
sk to appear in Sxe , if A sk +1 changes due to the action of procedures working for
other requirements, then Pke redefines ΨA (x) with same value and use.
The construction runs Pee or each e.
The verification follows the basic plan.
Lemma 3.1. If k > 0, then a single run of a procedure Pke calls at most g(δke )
e
many procedures Pk−1
.
e
Proof. Let t1 , t2 , . . . , tm be the stages at which a Pk−1
procedure returns, with
e
witnesses x1 , x2 , . . . xm , but the run of Pk does not return, necessarily because
e
which
c(rl , tl ) ≥ δke , where rl = ΨAtl (xl ) is the stage at which the run of Pk−1
returns at tl has defined ΨA (xl ). Since rl > tl−1 , the intervals in
{[rl , tl ) : l = 1, 2, . . . , m}
are pairwise disjoint, so m < g(δke ).
Lemma 3.2. For each k ≤ e, at most
nek
runs of
Pke
are ever called.
Proof. This is proved by reverse induction on k. For k = e this is clear; the
induction step uses Lemma 3.2, noting that we defined nek−1 = nek g(δke ).
Corollary 3.3. A run of P0e can always choose a fresh x ∈ Ie as a witness.
Lemma 3.4. If a run of Pke returns with a witness x at some stage, then |Sxe | ≥ k
at that stage.
Proof. This is proved by (forward) induction on k. It is clear for k = 0. Suppose
that this holds for k −1. Suppose that a run of Pke returns at some stage sk+1 . Then
e
there was a stage sk at which a procedure Pk−1
, called by this run of Pke , returned
with the same witness x, and by induction, at that stage sk , we had |Sxe | ≥ k − 1.
The run of Pke then defined ΨA (x) = sk . Note that sk is not in Sxe at stage sk . On
the other hand, sk appears in Sxe at stage sk+1 , so at that later stage we must have
|Sxe | ≥ k.
Lemma 3.5. Each requirement Re is met.
Proof. Lemma 3.4 implies that the original run of Pee cannot return. By induction
we see that there must be some k ≤ e such that some run of Pke never returns but,
e
if k > 0, every run of Pk−1
which is called by that run of Pke does return (we can
call that a golden run of Pke ).
e
By Lemma 3.2, there is a last run of Pk−1
which is called by the golden run of
Pke , and returns with witness x at some stage sk . (If k = 0 then s0 is the stage at
which the golden run of P0e is called, and x is the witness which is chosen.) The
golden run of Pke goes on to define ΨA (x) = sk and waits for ever for sk to show
up in Sxe . At the end of times we have ΨA (x) = sk so hSxe ix is not a trace of ΨA ,
which means that Re is met.
Lemma 3.6. The enumeration hAs i obeys c.
14
NOAM GREENBERG AND ANDRÉ NIES
Proof. For every e and every k = 1, 2, . . . , e, at most nek runs of Pke ever return, and
each time one such run returns, it enumerates into A a number whose cost at the
time is bounded by δke . Hence the total amount spent by all the runs of Pke is
nek δke = 2−e .
It follows that
X
s<ω
c(x, s)[[x least such that As (x) 6= As+1 (x)]] ≤
X
e2−e
e
which is finite.
Corollary 3.7 (Ng [12]). For every order function h there is an order function e
h
such that there is a c.e. set which is h jump-traceable but is not e
h jump-traceable.
Hence, there is no order function h such that the strongly jump-traceable degrees
coincide with the h jump-traceable degrees.
As mentioned in the introduction, it follows that the index-set of the strongly
jump-traceable sets is not Σ03 .
Proof. Because of the proximity between the tracing of all A-partial computable
functions and tracing J A , it is sufficient to show that for every order function h
there is a c.e. set A which is not strongly jump-traceable, but such that J A has a
c.e. trace bounded by h.
Given an order function h, by Proposition 2.2, let c be a benign cost function
such that for every c.e. set A which obeys c, J A has a c.e. trace bounded by h.
By Theorem 1.3, there is a c.e. set A which obeys c and is not strongly jumptraceable.
4. The diamond class for being LR-hard
In this section we prove Theorem 1.4: every strongly jump-traceable c.e. set is
computable in every LR-hard random set. The class LRH is Σ03 by the equivalence
of LR and LK-reducibility (see [15, 8.5.12]). Nonetheless, it is somewhat hard to
work with. We actually show that SJT is a subclass of a H♦ , where H is a class
which contains LRH and is nicer than LRH. The class H we use is the class of
∅0 -tracing sets.
Definition 4.1. A set X is ∅0 -tracing if there is some order function h such that
every ∆02 function f has an X-c.e. trace which is bounded by h.
Importantly, this definition is not a true relativisation of the notion of c.e. traceability. If it was, we would say that ∅0 is c.e. traceable relative to X if there is some
X-computable non-decreasing, unbounded function h such that every f ≤T ∅0 ⊕ X
has an X-c.e. trace bounded by h. Full relativisation of c.e. traceability, and in
fact of many other notions, does not yield useful concepts, at least not as useful
of partial relativisation as in Definition 4.1. It seems, though, that all appropriate
prepositions, such as ‘over’ and ‘in’, were already used by various authors to denote
full relativisation. Because of that (or possibly because of not being native English
speakers) the authors of this paper will instead use the new preposition “plop” to
indicate partial relativisation as in Definition 4.1. So a set X is ∅0 -tracing iff ∅0 is
c.e. traceable plop X.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
15
Proposition 4.2. Every set X ∈ LRH is ∅0 -tracing.
Proof. Even more is true: Simpson [18], relying on work of Kjos-Hanssen, Miller
and Solomon, extended the fact that every K-trivial is jump-traceable, and showed
that every X ∈ LRH is JT -hard, which means that ∅0 is jump-traceable plop X:
0
there is an order function h such that J ∅ has an X-c.e. trace bounded by h.
Now, the plan is to find some ∆02 function f and a benign cost function c such
that if A obeys c and X is a random set which traces f , then A ≤T X. Let h be
an order function. We can then fix a universal oracle trace for h:
c.e.
a uniformly
sequence hVn i of operators, such that for every oracle X ∈ 2ω , VnX is an X-c.e.
trace
function f which has an X-c.e. trace bounded
√ bounded by h, such that
every
by h is almost traced by VnX .
Given h, and consequently hVn i, we are interested in functions f such that for
all n, the measure of
Y : f (n) ∈ VnY
is at most 2−n . For the rest of this section,
we
will call such functions f rarely
traced for h. Namely, for few oracles Y is VnY a trace for f .
Lemma 4.3. Suppose h is an order function and f ≤T ∅0 is rarely traced for h.
Then there is a cost function c such that
A ≤T Y for every c.e. set A which obeys
c, and every random set Y such that VnY almost traces f . If f is also ω-c.e. then
c is benign.
Lemma 4.4. For every order function h, there is an ω-c.e. function f which is
rarely traced for h.
Proof of Theorem 1.4, given Lemmas 4.3 and 4.4. Let A be a strongly jump-traceable
c.e. set, and let Y be an LR-hard random set. By Proposition 4.2, Y is ∅0 -tracing.
Let e
h be an order function such that every ∆02 function has a Y -c.e. trace bounded
by e
h.
Leth = e
h2 . Then letting hVn i be the universal oracle trace for h, we know
Y
that Vn almost traces every ∆02 function.
By Lemma 4.4, there is an ω-c.e. function f which is rarely traced for h. By
Lemma 4.3, there is a benign cost function c such that if A is a c.e. set which obeys
c, then A ≤T Y . By the main Theorem 1.2, A obeys c.
Before we continue, we show the following:
Proposition 4.5. The ideal LRH♦ properly contains SJT.
Proof. As mentioned in the proof of 4.2, if X ∈ LRH, then ∅0 is jump-traceable plop
X. In fact, Simpson [18] showed that there is a there is a single order function h
such that for every X ∈ LRH, J X is c.e. traceable with bound h, and so that
there
is a single cost function h such that every ∆02 set is almost traced by VnX , where
hVn i is the universal oracle trace for h. The proof of Theorem 1.4 now holds with a
single cost function c which is obtained by applying Lemma 4.3 to a single function
given by Lemma 4.4. By Theorem 1.3, there is a c.e. set which obeys that cost
function but is not strongly jump-traceable.
This proof does not hold for the potentially smaller class (∅0 -tracing)♦ ; of course,
the proof above of Theorem 1.4 shows that
SJT ⊆ (∅0 -tracing)♦ ,
16
NOAM GREENBERG AND ANDRÉ NIES
but it is possible that these ideals coincide.
We turn to prove Lemmas 4.3 and 4.4. The proof of the latter uses ideas of
Hirschfeldt and the following measure theoretic version of the pigeon hole principle.
Fact 4.6. Let ε > 0 and k < ω. Let B be a collection of measurable subsets of 2ω
which has size greater than k/ε, such that every B ∈ B has measure at least ε.
Then there is some C ⊆ B of size k + 1 such that
\
C
is non empty (indeed, is not null).
Proof. [15, Ex. 1.9.15] Suppose that the intersection of any k + 1 sets in B is null.
Let
X
1B ,
f=
B∈B
where 1B is the indicator
function of B. The assumption implies that
R
R on a co-null
set, f (x) ≤ k, so f (x) dx ≤ k. On the other hand, for every B ∈ B, 1B (x) dx ≥ ε
and so
Z
XZ
k
f (x) dx =
1B (x) dx > ε = k,
ε
B∈B
which is a contradiction.
Proof of Lemma 4.4. Let h be an order function and let hVn i be the associated
universal oracle trace. We define an increasing approximation hfs i for the function
f , starting with f0 (n) = 0 for all n.
The idea is to keep, for each n, the measure of
Y
(1)
Y : fs (n) ∈ Vn,s
not greater than 2−n . So if we see at stage s that the measure of
Y
Y : fs−1 (n) ∈ Vn,s
is exceeding 2−n , then we redefine fs (n) = s, to keep the measure of the set (1)
bounded by 2−n at stage s.
To show that f is ω-c.e. (and well-defined), we see that for each n, the value
fs (n) changes at most 2n h(n) many times. For if m = fs−1 (n) and fs (n) = s, then
the set
n
Bm
= Y : m ∈ VnY
n
has measure at least 2−n . The intersection of Bm
for more than h(n) many such
Y
m’s is empty, because Vn ≤ h(n) for every Y . Fact 4.6 implies that there can be
no more than h(n)/2−n many m’s which are discarded as values for fs (n).
Proof of Lemma 4.3. Let h be an order function, hVn i the associated universal oracle trace, and let hfs (n)i be a computable approximation of a function f which is
rarely traced for h. By speeding up the approximation of f , we may assume that
at every stage s, the measure of
Y
Y : fs (n) ∈ Vn,s
is bounded by 2−n .
We first explain
the main ideas behind the proof. Suppose that Y is a random
set and that VnY almost traces f , and that A is a c.e. set which will obey the cost
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
17
function c that we will define. We want to show that Y computes A; suppose that
at stage s, for every n, we have committed that Y u(n) computes As k(n) , where
u(n) is a use marker and k(n) is a target marker. The main challenge, of course, is
to ensure the correctness of the computation once A k(n) changes. The mechanism
for doing that is enumerating a sequence of c.e. open classes hUn i, where Y ∈ Un
will roughly imply that Y u(n) has committed to compute a wrong initial segment
of A. We will
P ensure that the sets hUn i together add up to a Solovay test, in the
sense that n µ(Un ) is finite; we would then know that Y can be in at most finitely
many Un ’s, and so (from some point) must compute
P A correctly.
The challenge then moves to
make
the
sum
n µ(Un ) finite. This is where
we utilise the assumption that VnY traces f . A naı̈ve strategy for limiting the
Y
damage is setting k(n) = n, waiting for fs (n) ∈ Vn,s
, with some use u(n) and then
mapping Y u(n) to As n . Since the measure of the set of Z’s which trace fs (n)
in VnZ is at most 2−n , it would seem that this ensures that the measure of Un too
is bounded by 2−n . This, however, does not take into account the fact that ft (n)
may change after stage t; following the naı̈ve strategy would have us enumerate
a weight of 2−n for each possible value of ft (n), and this does not have a finite
total. The problem becomes acute when the following sequence of events occurs:
we have three approximations for f (n), at stage s1 , s2 and s3 . At stages t1 and
t3 (s1 < t1 < s2 < s3 < t3 ), the current value of f (n) is traced in VnY and so Y
computes both At1 n and At3 n . However, these are distinct, because a number
below n entered A between stages s2 and s3 (and fs2 (n) does not appear in VnY ).
The cost function c that we build can extract a “payment” from A only when A
changes, so that single change should not allow us to enumerate Y into Un .
The solution is to share the responsibility down the ladder. First, instead of just
Y
, we wait for fs (m), for all m ≤ n, to appear in
waiting for fs (n) to appear in Vn,s
Y
Y
Vm,s (or, if Vn only traces f from some m∗ onwards, all m ∈ [m∗ , n]). Then,
at any stage at which ft (m) changes, we let Y u(m) be responsible for computing
A n ; that is, we set k(m) ≥ n. This will correspond to setting c(n, t) ≥ 2−m . Then,
if A n changes, then we look at the least m such that k(m) ≥ n. This is the least
m such that the approximation for f (m) has changed since stage s. It follows that
the approximation for f (m − 1) has not changed since stage s, and so the current
Y
value of f (m − 1) is in Vm−1,s
. This allows us to enumerate Y into Um , and thus
record the change in A n .
We can now give the formal details. First, we define the cost function c and
show that it is benign.
For all x < ω, let c(x, 0) = 2−x . For any stage s > 0, let yt be the least number
y such that fs (y) 6= fs−1 (y). We let, for all x < s,
c(x, s) = max{c(x, s − 1), 2−yt },
(and leave c(x, s) unchanged for all x ≥ s). A short examination will reveal that
c is monotone and satisfies the limit condition. If hfs i is an ω-c.e. approximation
– say the mind-change function is bounded by a computable function g – then c
is benign: suppose that I is a set of pairwise disjoint intervals of natural numbers
such that for all [x, s) ∈ I we have c(x, s) ≥ 2−n . If [x, s) ∈ I and x > n, then
18
NOAM GREENBERG AND ANDRÉ NIES
fr (m) 6= fr−1 (m) for some m ≤ n and r ∈ [x, s). Hence
X
|I| ≤ (n + 1) +
g(m),
m≤n
which is a computable bound.
Let hAs i be an enumeration of a c.e. set A which obeys c.
We introduce some notation: for any stage t ≥ n, let sn (t) be the least stage
s ≤ t such that for all r ∈ [s, t], fr (n) = fs (n). By changing fn (n), we may assume
that for all t,
fsn (t) (n) 6= fsn (t)−1 (n),
so sn (t) ≥ n for all t ≥ n.
At stage t, let n be the least such that
At sn (t) 6= At−1 sn (t) .
For that n, enumerate all the sets Y such that
Y
ft (n − 1) ∈ Vn−1,t
into Un .
We note that by our assumption that the approximation for f is a “rarely traced”
one, that the measure of the collection of sets which are enumerated into Un at a
given stage is at most 2−(n−1) . Let nt be the unique number n such that sets are
enumerated into Y at stage t (let nt = ∞ is there is no such n). So
X
X
µ(Un ) ≤ 2
2−nt .
n
t
Secondly, if sets are enumerated into Un at a stage t, then there is some x <
sn (t) such that At (x) 6= At−1 (x), and such that c(x, t) ≥ 2−n . So letting mt =
− log2 c(xt , s), where xt is the least number x such that At (x) 6= At−1 (x), we get
mt ≤ nt for all t, so
X
X
2−nt ≤
2−mt .
t
t
The
P assumption that hAs i obeys c means that the sum on the right is finite. Hence
n µ(Un ) is finite as well.
By standard randomness arguments (for example, thinking of the union of the
Un ’s as a Solovay test), if Y is random,
/ Un for almost all n. Let Y be
then Y ∈
a random set, and suppose that VnY almost traces f . To complete the proof, we
need to show that A ≤T Y . Let n∗ be large enough so that:
• Y ∈
/ Un for all n > n∗ , and
• f (n) ∈ VnY for all n ≥ n∗ .
Let s∗ be a stage late enough so that
fs n∗ +1 = f n∗ +1
∗
for all s ≥ s , and such that no sets are enumerated into any Um for m ≤ n∗ after
stage s∗ .
Let n > n∗ . We claim that:
Y
If t > s∗ and for all m ∈ [n∗ , n) we have ft (m) ∈ Vm,t
, then
At sn (t) = A sn (t) .
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
19
For suppose, for contradiction, that there is a stage u > t such that Au s 6= Au−1 s ,
where s = sn (t). Let m be the least such that Au sm (u) 6= Au−1 sm (u) . Since u > s∗ ,
we have m > n∗ ; but also m ≤ n because sn (u) ≥ sn (t) = s.
Y
Y
Now we claim that fu (m − 1) ∈ Vm−1,u
. For by assumption, ft (m − 1) ∈ Vm−1,t
,
Y
the latter set is a subset of Vm−1,u , and if fu (m − 1) 6= ft (m − 1) then sm−1 (u) >
t ≥ s, which would contradict the minimality of u. Hence Y gets enumerated into
Um at stage u, which is a contradiction.
Now as sn (t) ≥ n for all n, we certainly have A ≤T Y .
5. (ω-c.e.)♦ and Superlow♦
We show that SJT ⊆ (ω-c.e.)♦ (Theorem 1.5) and that Superlow♦ is properly
contained in the K-trivial degrees (Theorem 5.3). Of course, Superlow ⊆ ω-c.e. and
so (ω-c.e.)♦ ⊆ Superlow♦ .
Theorem 1.5 is an immediate consequence of the main Theorem 1.2 and the
following proposition.
Proposition 5.1. Let Y be a ∆02 random set. Then there is a cost function c such
that every c.e. set which obeys c is computable from Y . If, further, Y is ω-c.e.,
then c is benign.
Proof. This is similar to the proof of Lemma 4.3. Let hYs i be a computable approximation for Y . The idea is to let Y n−1 compute A s if Ys n 6= Ys−1 n . So let
c(x, 0) = 2−x , and for s > 0, if n = ns is the least such that Ys n 6= Ys−1 n , then
we let, for all x < s,
c(x, s) = max{c(x, s − 1), 2−n }.
Again it is easy to verify that c is monotone (and satisfies the limit condition). If,
further, the number of stages s such that Ys (m) 6= Ys−1 (m) is bounded by g(m),
where g is a computable function, and I is a set of pairwise disjoint intervals of
natural numbers such that for all [x, s) ∈ I we have c(x, s) ≥ 2−n , then for all
[x, s) ∈ I such that x > n, there is some t ∈ (x, s] such that Yt n 6= Yt−1 n , so
X
|I| ≤ (n + 1) +
g(m).
m<n
Thus if hYs i is an ω-c.e. approximation for Y , then c is benign.
Now suppose that hAs i is an enumeration of a c.e. set A which obeys c. For all n
and t ≥ n, let sn (t) be the least stage s ≤ t such that for all r ∈ [s, t], Yr n = Yt n .
Again, without loss of generality, sn (t) ≥ n for all t ≥ n. At a stage t, if n is least
such that At sn (t) 6= At−1 sn (t) , then we enumerate Yt n−1 into a Solovay test G.
P
The fact that hAs i obeys c implies that indeed, the sum σ∈G 2−|σ| is finite, with
an argument which mirrors the one in the proof of Lemma 4.3.
Since Y is random, only finitely many initial segments of Y are enumerated into
G; suppose that the last one is enumerated at some stage s∗ . We now claim that if
t > s∗ and
Y n−1 = Yt n−1 ,
then
A sn (t) = At sn (t) .
For otherwise, let u > t be a stage at which Au sn (t) 6= Au−1 sn (t) ; let m be the
least such that Au sm (u) 6= Au−1 sm (u) . We know that m > n∗ because u > s∗ , and
m ≤ n because sn (u) ≥ sn (t).
20
NOAM GREENBERG AND ANDRÉ NIES
At stage u, we enumerate Yu m−1 into G. By minimality of m, we have
sm−1 (u) < s ≤ t, which implies that Yu m−1 = Yt m−1 ⊆ Yt n−1 which is an
initial segment of Y . This contradicts u > s∗ .
In fact, we can improve Proposition 5.1 to show that the use of the reduction
of A to Y can grow as slowly as we like. Already we see that the proof gives us
A ≤wtt Y , indeed A ≤ibT Y : A is reducible to Y with the use of the reduction
bounded by the identity function. In fact, we can have the use grow as slowly as
we like.
Proposition 5.2. If A is a strongly jump-traceable c.e. set and Y is an ω-c.e.
random set, then for every order function h there is a Turing reduction of A to Y
whose associated use function is bounded by h.
In the terminology of [8], A ≤T (tu) Y : A is reducible to Y with tiny use.
Proof. The formulation which we prove is: if h is a strictly increasing recursive
function, then there is a Turing functional Φ such that Φ(Y ) = A and such that
for all n, |Φ(Y n )| ≥ h(n).
We modify the proof of Proposition 5.1. Let h be an increasing recursive function.
Define, for all x,
c(x, 0) = max 2−n : h(n) ≤ x
and for s > 0, if n is the least such that Ys n 6= Ys−1 n , let, for all x < h(s),
c(x, s) = max c(x, s − 1), 2−n .
The argument in the proof of Proposition 5.1 shows that if the number of stages s
such that Ys (m) 6= Ys−1 (m) is bounded by a computable function g, and I is a set
of pairwise disjoint intervals of numbers such that for all [x, s) ∈ I, c(x, s) ≥ 2−n ,
then
X
|I| ≤ h(n) + 1 +
g(m),
m≤n
so c is benign.
Say that hAs i obeys c. Again we let, for all n and t ≥ h(n), sn (t) be the least
stage s ≤ t such that for all r ∈ [s, t] we have Yr n = Yt n ; by manipulating
the approximation hYs i, we may assume that for all n, for all t ≥ h(n), we have
sn (t) ≥ h(n). The rest of the proof now follows the proof of Proposition 5.1
verbatim, to show that for almost all t, if Y n−1 = Yt n−1 , then A sn (t) = At sn (t) .
This gives us a Turing functional Φ such that Φ(Y ) = A and such that for all n,
|Φ(Y n )| ≥ h(n) as required.
We turn to prove:
Theorem 5.3. There is a K-trivial degree which is not in Superlow♦ .
Theorem 5.3 follows from Proposition 5.4, applied to any Π01 class which only
contains random sets.
Proposition 5.4. If P is a non-empty Π01 class, then there is some superlow Y ∈ P
and a K-trivial c.e. set B such that B is not computable from Y .
Proof. This is an elaboration on the (super)low basis theorem. At stage s, we define
a sequence hP0,s , P1,s , . . . , P2s,s i as follows.
• Let P0,s = P.
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES
21
• Given P2e,s , if we see at stage s that J X (e)↓ for all X ∈ P2e,s , then we let
P2e+1,s = P2e,s ; otherwise, we let
P2e+1,s = {X ∈ P2e,s : J X (e)↑}.
• Given P2e+1,s , we try to meet the requirement Re which states that Φe (Y ) 6=
B. Each such requirement will choose a witness xe . If we see, at stage s,
that for all X ∈ P2e+1,s we have Φe (X, xe ) ↓= 0,then we let P2e+2,s =
P2e+1,s (and enumerate xe into B if not done so already). Otherwise, we
let
P2e+2,s = {X ∈ P2e+1,s : Φe (X, xe )↑ ∨ Φe (X, xe )↓= 1}.
At the beginning of the next stage, witnesses for the requirements Re are updated:
if P2e+1,s 6= P2e+1,s−1 , then we pick a fresh witness for Re . At stage s, let is (e) be
the number of stages s at which P2e+1,s 6= P2e+1,s−1 . If cK (xe , s + 1) > 2−(e+is (e)) ,
then we pick a new, fresh witness for Re . Here again cK is the standard cost
function for K-triviality.
Now by induction, we can show that the Π01 classes Pk reach a limit, and indeed
there is a computable bound on the number of changes of each Pk . As usual, after
P2e,s has stabilised, P2e+1,s changes at most once. So if there are at most n2e
versions of P2e,s , then there are at most n2e+1 = 2n2e versions of P2e+1,s .
As long as P2e+1,s does not change, xe can change at most 2e+is (e) many times
(as cK is benign with bound 2n ). Beyond those changes, P2e+2,s can change at
most once before it is injured by a change in P2e+1,s . So there are at most
X
n2e+2 =
2e+i + 1
i≤n2e+1
changes in P2e+2,sT
. The map k 7→ nk is computable.
It follows that Pk is a singleton {Y }, that Y is superlow, and that Y does not
compute B. Since B obeys cK , it is K-trivial.
We finish with an application to c.e. degree theory. We first need a plopification
– again, more than a mere relativisation – of the superlow basis theorem.
Proposition 5.5. Let P be a non-empty Π01 class, and let B ∈ 2ω . Then there is
some Y ∈ P such that
(Y ⊕ B)0 ≤tt B 0 .
Proof. For a string τ , let
Qτ = Y ∈ P : ∀ e < |τ | τ (e) = 0 → J Y ⊕B (e)↑ .
Note that Qτ is a Π01 (B) class, uniformly in τ . Emptyness of such a class is a Σ01 (B)
condition, so there is a computable function g such that
Qτ = ∅ ↔ g(τ ) ∈ B 0 .
Thus, there is a Turing functional Ψ such that ΨX is total for each oracle X, and
Ψ(B 0 , e) = τe , whereTτe is the leftmost string τ of length e + 1 such that Qτ is
non-empty. Let Y ∈ e Qτe . Then
e ∈ (Y ⊕ B)0 ↔ τe (e) = 1,
so (Y ⊕ B)0 ≤tt B 0 .
22
NOAM GREENBERG AND ANDRÉ NIES
Corollary 5.6 (Ng [13]). There is an almost superdeep c.e. degree: a c.e. degree
a such that for every superlow c.e. degree b, a ∨ b is also superlow.
Indeed, every strongly jump-traceable c.e. degree is almost superdeep.
Proof. Let A be a strongly jump-traceable c.e. set, and let B be a superlow set (in
fact we don’t need the hypothesis that B is c.e.). By Proposition 5.5 applied to a
Π01 class of random sets, there is a random set Y such that
Y 0 ≤tt (Y ⊕ B)0 ≤tt B 0 ≤tt ∅0 ,
so Y is superlow. By Theorem 1.5, we have A ≤T Y , so
(A ⊕ B)0 ≤tt (Y ⊕ B)0 ≤tt ∅0 ,
so A ⊕ B is superlow as well.
Corollary 5.7 (Diamondstone [5]). There is a promptly simple c.e. degree which
does not join to ∅0 with any superlow c.e. set.
Proof. Certainly if a degree is almost superdeep, then it does not join to ∅0 with any
superlow c.e. set. In [7], the authors show that there is a strongly jump-traceable
c.e. set which is promptly simple. By our proof of Corollary 5.6, such a set is almost
superdeep.
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